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1.6
(a)
Consider the following statistical model:
i
i
i
l
y
ε
τ
+
+
=
,
i
=1,2,…,6,
where
y
i
=the
i
th
observed difference between A and B (AB)
τ
=the intrinsic difference between A and B (AB)
l
i
=learning effect of the
i
th
transcript
ε
i
=errors with mean 0
When the test sequence for the
i
th
transcript is AB, B is benefited by the
learning effect, thus
l
i
<0.
Similarly,
l
i
>0 if the sequence of the
i
th
test is BA.
Assume that
0
...
6
2
1
>
=
=
=
=
l
l
l
l
in part (a).
Without randomization, as the following sequence:
AB, AB, AB, AB, AB, AB
l
y
−
=
=
η
ˆ
, i.e., the estimation of the difference between A and B is biased by
l
.
With randomization as the following sequence:
AB, BA, AB, BA, AB, AB
3
6
2
4
ˆ
l
l
l
y
−
=
−
−
=
=
, i.e., the estimation of the difference between A and B
is biased by
l
/3<
l
.
Using balance in addition to randomization, as the following sequence
AB, AB, AB, BA, BA, BA
(*)
=
−
−
=
=
6
3
3
ˆ
l
l
y
, i.e., there is no bias if we use balance.
(b)
I would not use the sequence (*).
A better choice is as follows:
AB, BA, BA, AB, AB, BA
(**)
For the sequence (*), all the 3 ABs are before all the BAs, which makes this
sequence uneasy.
Both sequence (*) and (**) do not cause estimation bias if
l
l
l
l
=
=
=
=
6
2
1
...
. However, for
0
...
6
2
1
>
>
>
>
l
l
l
, (**) leads to smaller bias
than (*).
1.10
a)
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View Full Document BodyWt
BrainWt
7000
6000
5000
4000
3000
2000
1000
0
6000
5000
4000
3000
2000
1000
0
Scatterplot of BrainWt vs BodyWt
The scatter plot above is not very informative. Due to the large values of Brain Weight
and Body Weight for a few observations, the abscissa and ordinate axes have to cover
very wide ranges. However, most of the observations are associated with small Brain
Weight and Body Weight. Consequently, many points cluster around the bottom left
corner of the scatter plot.
BodyWt
600
500
400
300
200
100
0
800
700
600
500
400
300
200
100
0
Scatterplot of BrainWt vs BodyWt
The above figure is a scatter plot of Brain Weight versus Body Weight with reduced
range for the abscissa and ordinate axes. Note that several observations are omitted from
the plot. However, it can now be seen that Brain Weight and Body Weight are positively
correlated.
b)
Log(BodyWt)
Log(BrainWt)
10.0
7.5
5.0
2.5
0.0
2.5
5.0
10
8
6
4
2
0
2
Scatterplot of Log(BrainWt) vs Log(BodyWt)
A scatter of the natural logarithm of Brain Weight versus the natural logarithm of Body
Weight is shown above. It is clear from the plot that there is a linear relationship between
the logarithm of Brain Weight and the logarithm of Body Weight. Taking the logarithm
transformation on both variables has the remarkable effect of introducing almost perfect
linearity in the relationship between the variables.
c)
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This note was uploaded on 12/25/2011 for the course ISYE 6413 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Staff

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